Comparison of Annual Returns

Oct 13, 2025 by wbrum.org 97 views

Introduction

When it comes to investing, understanding the interest rates and compounding methods is crucial in determining the annual returns. In this article, we will compare the annual returns of two individuals, Zahra and Christopher, who invested the same amount in different accounts with varying interest rates and compounding methods.

Zahra's Investment

Zahra invested $8,800 in an account paying an interest rate of $8 \frac{5}{8} \%$ compounded annually. To calculate the annual return, we need to convert the mixed fraction to a decimal.

Converting Mixed Fraction to Decimal

To convert the mixed fraction $8 \frac{5}{8}$ to a decimal, we divide the numerator by the denominator.

\frac{5}{8} = 0.625

Now, we add the whole number part to the decimal part.

8 + 0.625 = 8.625\%

Calculating Annual Return

The formula to calculate the annual return is:

A = P(1 + r)^n

where:

$A$ is the amount after $n$ years
$P$ is the principal amount (initial investment)
$r$ is the annual interest rate (in decimal form)
$n$ is the number of years

In this case, the principal amount $P$ is $8,800, the annual interest rate $r$ is $8.625\%$ or $0.08625$ in decimal form, and the number of years $n$ is $1$ .

A = 8800(1 + 0.08625)^1

A = 8800(1.08625)

A = 9542

So, the annual return for Zahra's investment is $9542 - 8800 = 742$ .

Christopher's Investment

Christopher invested $8,800 in an account paying an interest rate of $8 \frac{1}{4} \%$ compounded annually. To calculate the annual return, we need to convert the mixed fraction to a decimal.

Converting Mixed Fraction to Decimal

To convert the mixed fraction $8 \frac{1}{4}$ to a decimal, we divide the numerator by the denominator.

\frac{1}{4} = 0.25

Now, we add the whole number part to the decimal part.

8 + 0.25 = 8.25\%

Calculating Annual Return

Using the same formula as before, we can calculate the annual return for Christopher's investment.

A = P(1 + r)^n

where:

$A$ is the amount after $n$ years
$P$ is the principal amount (initial investment)
$r$ is the annual interest rate (in decimal form)
$n$ is the number of years

In this case, the principal amount $P$ is $8,800, the annual interest rate $r$ is $8.25\%$ or $0.0825$ in decimal form, and the number of years $n$ is $1$ .

A = 8800(1 + 0.0825)^1

A = 8800(1.0825)

A = 9492

So, the annual return for Christopher's investment is $9492 - 8800 = 692$ .

Comparison of Annual Returns

Now that we have calculated the annual returns for both Zahra and Christopher, we can compare the results.

Investor	Annual Return
Zahra	$742
Christopher	$692

As we can see, Zahra's investment yielded a higher annual return of $742 compared to Christopher's investment of $692. This is because Zahra's account had a higher interest rate of $8 \frac{5}{8} \%$ compared to Christopher's account with an interest rate of $8 \frac{1}{4} \%$ .

Conclusion

Q: What is the main difference between Zahra and Christopher's investments?

A: The main difference between Zahra and Christopher's investments is the interest rate. Zahra's account had an interest rate of $8 \frac{5}{8} \%$ , while Christopher's account had an interest rate of $8 \frac{1}{4} \%$ .

Q: Why did Zahra's investment yield a higher annual return?

A: Zahra's investment yielded a higher annual return because her account had a higher interest rate of $8 \frac{5}{8} \%$ compared to Christopher's account with an interest rate of $8 \frac{1}{4} \%$ .

Q: What is the formula to calculate the annual return?

A: The formula to calculate the annual return is:

A = P(1 + r)^n

where:

$A$ is the amount after $n$ years
$P$ is the principal amount (initial investment)
$r$ is the annual interest rate (in decimal form)
$n$ is the number of years

Q: How do I convert a mixed fraction to a decimal?

A: To convert a mixed fraction to a decimal, you need to divide the numerator by the denominator and then add the whole number part to the decimal part.

Q: What is the difference between simple interest and compound interest?

A: Simple interest is calculated as a percentage of the principal amount, while compound interest is calculated as a percentage of the principal amount plus any accrued interest.

Q: Can I use the same formula to calculate the annual return for both simple and compound interest?

A: No, you cannot use the same formula to calculate the annual return for both simple and compound interest. The formula for compound interest is:

A = P(1 + r)^n

while the formula for simple interest is:

A = P(1 + rt)

where:

$A$ is the amount after $t$ years
$P$ is the principal amount (initial investment)
$r$ is the annual interest rate (in decimal form)
$t$ is the time period in years

Q: How do I determine the number of years for the investment?

A: The number of years for the investment is usually specified in the investment contract or agreement. In this case, we assumed that both Zahra and Christopher invested for 1 year.

Q: Can I use this formula to calculate the annual return for any type of investment?

A: No, this formula is specifically designed for calculating the annual return for investments with a fixed interest rate and a specified time period. You may need to use a different formula or approach for other types of investments, such as stocks or mutual funds.

Q: What are some common mistakes to avoid when calculating the annual return?

A: Some common mistakes to avoid when calculating the annual return include:

Using the wrong formula or approach for the type of investment
Failing to convert mixed fractions to decimals
Using incorrect values for the principal amount, interest rate, or time period
Failing to account for compounding or other factors that may affect the investment

Q: How can I improve my understanding of investments and calculate the annual return accurately?

A: To improve your understanding of investments and calculate the annual return accurately, you can:

Read and understand the investment contract or agreement
Research and understand the different types of investments and their characteristics
Use online calculators or software to help with calculations
Seek advice from a financial advisor or investment professional
Practice calculating the annual return for different scenarios and investments